On the inviscid neutral curve of rotating Poiseuille pipe flow

Abstract

This paper concerns linear instabilities of incompressible flow along a rotating pipe when the disturbances are inviscid. The inviscid spectrum comprises an infinite number of eigenmodes, and significant complications arise from (i) switching in the identity of the most unstable mode and (ii) the existence of distinguished parameter limits in which some or all of the modes acquire special scalings. Nevertheless, we show that in one sense all the instabilities form a single ordered family, which we term a center-mode family, by applying an asymptotic theory recently developed for the trailing line vortex flow. We give complete computed neutral curves (stability boundaries) over a wide range of Rossby numbers, as well as contour plots of maximum growth rate. We find that the numerical calculations are in good agreement with the asymptotic theory and correctly recover previous results, including the limits of very large and very small Rossby number. There is some disagreement, however, with published results on the neutral curve at finite Rossby number. We show that this disagreement results from the intricate nature of the spectrum, which possesses two different types of neutral mode with two distinguished scalings.